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A STOCHASTIC MARKOV CHAIN
MODEL TO DESCRIBE CANCER
METASTASIS
by
Jeremy Mason
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements of the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
18 December 2013
Copyright 2013 Jeremy Mason

A stochastic Markov chain model for metastatic progression is developed for primary 8 major cancer types based on a network construction of metastatic sites with dynamics modeled as an ensemble of random walkers on the network. We calculate a transition matrix for each primary cancer and use it to construct a circular bi-directional network of primary and metastatic locations based on postmortem tissue analysis of 3827 autopsies on untreated patients documenting all primary tumor locations and metastatic sites from this population. The resulting 50 potential metastatic sites are connected by directed edges with distributed weightings, where the site connections and weightings are obtained by calculating the entries of an ensemble of transition matrices so that the steady-state distribution obtained from the long-time limit of the Markov chain dynamical system corresponds to the ensemble metastatic distribution obtained from the autopsy dataset. We condition our search for a transition matrix on an initial distribution of metastatic tumors obtained from the dataset. Through an iterative numerical search procedure, we adjust the entries of a sequence of approximations until a transition matrix with the correct steady-state is found (up to a numerical threshold). Once the transition matrix for a given cancer type is computed, our metastatic progression model is based Monte Carlo simulations of collections of random walkers all leaving the primary tumor location and executing a random walk across the directed graph from site to site. The model allows us to simulate and quantify disease progression pathways and timescales of progression from the primary tumor location to other sites. Pathway diagrams are created that classify metastatic tumors as 'spreaders' or 'sponges' and quantif es three types of multidirectional mechanisms of progression: (i)self-seeding of the primary tumor, (ii) reseeding of the primary tumor from a metastatic site (primary reseeding), and (iii) reseeding of metastatic tumors (metastasis reseeding). The entire process is replicated for additional primary tumors in the dataset of for individual analysis and comparative purposes. ❧ A second contribution of this work is to introduce a quantitative notion of 'metastatic entropy' for cancer and use it to compare the complexity and predictability associated with the 12 most common cancer types worldwide. We apply these notions of entropy and predictability directly to the autopsy dataset used to create our Markov model. The raw data, which contains the number of metastases found at all of the anatomical sites for each cadaver (whose primary tumor location is also recorded), is normalized (yielding their empirical distribution) so that we can interpret the histograms as probability mass functions (PMFs) representing the large scale (whole body) metastatic 'signature' of each primary cancer. We characterize the power-law distributions associated with metastatic tumor distributions for each primary cancer type. Then we calculate the entropy associated with each and use the Kullback-Liebler divergence (relative-entropy) to compare each cancer type with all of the data aggregated into an 'all cancer' category, whose entropy value is used as a benchmark for comparisons.

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A STOCHASTIC MARKOV CHAIN
MODEL TO DESCRIBE CANCER
METASTASIS
by
Jeremy Mason
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements of the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
18 December 2013
Copyright 2013 Jeremy Mason